
functions of the form \(\sum _{i=1}^k\alpha _it^{b_i}\)  where \(\alpha _1,\ldots , \alpha _k\in {\mathbb {R}}\) , \(b_1,\ldots , b_k\in (0,+\infty )\) , and every non-trivial linear combination
of \(a_1,\ldots , a_\ell \) 
is not a rational polynomial (in [1]} this was established when \(b_1,\ldots , b_k\in {\mathbb {Z}}_+\) ).

linearly independent functions of the form \(\sum _{i=1}^k\alpha _i \, t^{b_i}(\log {t})^{c_i}\) , where \(b_i\in {\mathbb {R}}_+\setminus {\mathbb {Z}}_+\) , \(\alpha _i, c_i\in {\mathbb {R}}\) , \(i=1,\ldots , k\) .

